# Bear in mind that is a function of t or however k1 0

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Unformatted text preview: i=1 The formula (5.5.5a) has the general form of (5.1.2a) with k X i=0 i yn;i = h 0 fn 2 (tn;k tn): 1 = 2 = ::: = (5.5.5b) k = 0, i.e., (5.5.5c) The backward-di erence formula with index k is a k-step method having order k (since the local discretization error is O(hk ). Backward di erence methods and their local discretization errors for k = 1 2 3 4, follow. Backward di erence coe cients for k = 1 2 : : : 6, appear in Table 5.5.1 1]. 25 k = 1 : Backward Euler method yn = yn;1 + hfn (5.5.6a) = ; h y00 ( ): 2 (5.5.6b) n k = 2 : Two-step backward-di erence formula ryn + r2yn = hfn 2 or yn = 1 (4yn;1 ; yn;2 + 2hfn) 3 n 2 = ; h y000( ): 3 (5.5.7a) (5.5.7b) k = 3 : Three-step backward-di erence formula 1 yn = 11 (18yn;1 ; 9yn;2 + 2yn;3 + 6hfn) h iv n = ; y ( ): 4 3 (5.5.8a) (5.5.8b) k = 4 : Four-step backward-di erence formula 1 yn = 25 (48yn;1 ; 36yn;2 + 16yn;3 ; 3yn;4 + 12hfn) h4 yv ( ): n=; 5 (5.5.9a) (5.5.9b) Examples using backward-di erence formulas will be presented in Section 5.7. 26 k 0 0 1 2 3 4 5 6 1 1 1 1 -1 2 3 2 3 -4 1 3 11 6 11 -18 9 -2 4 25 12 25 -48 36 -16 3 5 137 60 137 -300 300 -200 75 -12 6 147 60 147 -360 450 -400 225 -72 10 Table 5.5.1: Coe cients of the backward di erence method (5.5.5c). for orders one through six 1]. Their local error coe cients are 1=(k + 1). 5.6 Convergence, Accuracy, and Stability Having several methods at our disposal, let us now study questions of consistency, convergence, and stability of linear multistep methods (LMMs) of the form (5.1.2) applied to the IVP (5.1.1). We begin by applying the usual de nitions in our present situation. De nition 5.6.1. Let z(t) 2 C 1 0 T ], tn ; ih 2 0 T ], i = 0 1 : : : k, and Pk k X0 i=0 i z (tn ; ih) ; L (z(t)) = z (t ; ih): h h i=0 i n (5.6.1a) The local discretization error of the LMM (5.1.2) is = Lh(y(tn)): (5.6.1b) dn = y(tn) ; yn (5.6.2) n De nition 5.6.2. The local error is assuming that no errors have occurred prior to tn, i.e., yi = y (ti), i < n. Substituting exact solution values into (5.1.2) for i = 0...
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